PlanetMath: variation of parameters

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variation of parameters

(Theorem)

The method of variation of parameters is a way of finding a particular solution to a nonhomogeneous linear differential equation.

Suppose that we have an th order linear differential operator

$\displaystyle L[y] := y^{(n)} + p_1(t) y^{(n-1)} + \cdots + p_n(t) y ,$

and a corresponding nonhomogeneous differential equation

$\displaystyle L[y] = g(t).$

(1)

Suppose that we know a fundamental set of solutions $y_1,y_2,\ldots,y_n$ of the corresponding homogeneous differential equation

. The general solution of the homogeneous equation is

$\displaystyle y_c(t) = c_1 y_1(t) + c_2 y_2(t) + \cdots + c_n y_n(t),$

where $c_1,c_2,\ldots,c_n$ are constants. The general solution to the nonhomogeneous equation

is then

$\displaystyle y(t) = y_c(t) + Y(t),$

where

is a particular solution which satisfies

, and the constants $c_1,c_2,\ldots,c_n$ are chosen to satisfy the appropriate boundary conditions or initial conditions.

The key step in using variation of parameters is to suppose that the particular solution is given by

$\displaystyle Y(t) = u_1(t) y_1(t) + u_2(t) y_2(t) + \cdots + u_n(t) y_n(t),$

(2)

where $u_1(t),u_2(t),\ldots,u_n(t)$ are as yet to be determined functions (hence the name variation of parameters). To find these

functions we need a set of

independent equations. One obvious condition is that the proposed ansatz satisfies Eq. (1). Many possible additional conditions are possible, we choose the ones that make further calculations easier. Consider the following set of

conditions

$\displaystyle y_1 u_1' + y_2 u_2' + \cdots + y_n u_n'$	$\displaystyle =$	0
$\displaystyle y_1' u_1' + y_2' u_2' + \cdots + y_n' u_n'$	$\displaystyle =$	0
		$\displaystyle \vdots$
$\displaystyle y_1^{(n-2)} u_1' + y_2^{(n-2)} u_2' + \cdots + y_n^{(n-2)} u_n'$	$\displaystyle =$	$\displaystyle 0.$

Now, substituting Eq. (2) into

and using the above conditions, we can get another equation

$\displaystyle y_1^{(n-1)} u_1' + y_2^{(n-1)} u_2' + \cdots + y_n^{(n-1)} u_n' = g .$

So we have a system of equations for $u_1',u_2',\ldots,u_n'$ which we can solve using Cramer's rule:

$\displaystyle u_m'(t) = \frac{g(t) W_m(t)}{W(t)}, \quad m=1,2,\ldots,n .$

Such a solution always exists since the Wronskian $W=W(y_1,y_2,\ldots,y_n)$ of the system is nowhere zero, because the $y_1,y_2,\ldots,y_n$ form a fundamental set of solutions. Lastly the term

is the Wronskian determinant with the

th column replaced by the column $(0,0,\ldots,0,1)$ .

Finally the particular solution can be written explicitly as

$\displaystyle Y(t) = \sum_{m=1}^n y_m(t) \int \frac{g(t) W_m(t)}{W(t)} dt .$

Bibliography

1: W. E. Boyce and R. C. DiPrima. Elementary Differential Equations and Boundary Value Problems John Wiley & Sons, 6th edition, 1997.

"variation of parameters" is owned by rspuzio. [ owner history (2) ]

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Other names:

variation of constants

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Cross-references: column, term, Wronskian, Cramer's rule, Ansatz, obvious, independent, functions, initial conditions, boundary conditions, equation, homogeneous equation, general solution, homogeneous differential equation, solutions, differential equation, differential operator, order, linear differential equation, nonhomogeneous, particular solution
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This is version 5 of variation of parameters, born on 2002-05-20, modified 2005-03-06.
Object id is 2921, canonical name is VariationOfParameters.
Accessed 14520 times total.

Classification:

AMS MSC:	34A05 (Ordinary differential equations :: General theory :: Explicit solutions and reductions)
	34A30 (Ordinary differential equations :: General theory :: Linear equations and systems, general)

Pending Errata and Addenda

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Discussion

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variation of parameters by mcintosh on 2004-02-08 17:34:19

I am not sure why this item lies in the "lacking a proof" list. Perhaps it is because of the title, which implies an ad-hoc method of solution. If one approaches this through matrix calculus, having discussed the solution of dZ/dt = MZ with a functional coefficient matrix M(t) and full matrix of solutions Z(t), the way to approach the inhomogeneous equation dW/dt = MW + F is to factor the solution matrix, writing W = ZQ.

Then dW/dt = dZ/dt Q + Z dQ/dt = MZQ + F. Supposing now that the homogeneous equation has already been solved, and therefore a known quantity, it remains to solve Z dQ/dt = F. A result from matrix calculus assures us that Z is invertible if it evolves from a basis of initial conditions, leaving the quadrature (that is, pure integration, no solving) Q = Q(0) + Integral(0,t)[Z^-1(s)F(s)ds].

Since the inverse of a matrix is its adjugate divided by its determinant (which in the case of an n'th order equation is a Wronskian), there results the nice formula in this posting.

Admittedly to convert the above into a proof, some details from matrix calculus have to be assumed, such as the existence of a solution in the form of the matrizant (Picard's method) or as a product integral (Euler's method). Likewise that d|Z|/dt = Trace(Z)|Z|, by having an exponential solution which never vanishes for analytic Z, guarantees the invertibility of Z.

- hvm

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